Successive Averages of the Vertices of a Polygon

If the vertices (thought of as vectors) of a polygon are , then the midpoints of the sides are . These new points form the vertices of another polygon, and the midpoints of its sides are . This process can be repeated for iterations.

There is another variable, , the number of successive points around the polygon that can be averaged. Here are the next two initial averages (and ):

The original seven points are scattered randomly. For clarity, drag them to form a convex or star-shaped polygon and experiment with the controls.